Infinite series
Objective. Distinguish sequence convergence from series convergence.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
An infinite series is the sum a₁ + a₂ + a₃ + … formed from a sequence, and it converges when the sequence of partial sums Sₙ = a₁ + … + aₙ tends to a finite limit, which is quite different from the underlying sequence (aₙ) itself converging to zero. A necessary (but not sufficient) condition for convergence is aₙ → 0, since if terms do not vanish the partial sums cannot settle. Geometric series Σ arⁿ converge exactly when |r| < 1, with sum a/(1 − r). The key distinction is that aₙ → 0 does not guarantee the series converges: the harmonic series Σ 1/n has terms tending to 0 yet its partial sums grow without bound, so the series diverges.
Worked example
Determine whether the infinite series Σ (from n=1 to ∞) of 3(1/4)ⁿ⁻¹ converges, and if so find its sum.
- 1Identify this as geometric with first term a = 3 and common ratio r = 1/4.
- 2Check convergence: since |r| = 1/4 < 1, the series converges.
- 3Apply the sum formula S = a/(1 − r) = 3/(1 − 1/4).
- 4Simplify: S = 3/(3/4) = 4.
Answer: 4
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Practice
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Start practisingCommon mistake. Students often claim a series converges because its terms tend to 0, as with Σ 1/n. Always check partial sum behaviour or use a proper test, since the condition aₙ → 0 is necessary but never sufficient alone.
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